Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{5}z^{3})^{-1}}}{{(a^{4}z^{-5})^{-1}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{5}z^{3})^{-1} = (a^{5})^{-1}(z^{3})^{-1}}$ On the left, we have ${a^{5}}$ to the exponent ${-1}$ . Now ${5 \times -1 = -5}$ , so ${(a^{5})^{-1} = a^{-5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{5}z^{3})^{-1}}}{{(a^{4}z^{-5})^{-1}}} = \dfrac{{a^{-5}z^{-3}}}{{a^{-4}z^{5}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-5}z^{-3}}}{{a^{-4}z^{5}}} = \dfrac{{a^{-5}}}{{a^{-4}}} \cdot \dfrac{{z^{-3}}}{{z^{5}}} = a^{{-5} - {(-4)}} \cdot z^{{-3} - {5}} = a^{-1}z^{-8}$